L1(R)-Nonlinear Stability of Nonlocalized Modulated Periodic Reaction-Diffusion Waves

Assuming spectral stability conditions of periodic reaction-diffusion waves u(x), we consider L1(R)-nonlinear stability of modulated periodic reaction-diffusion waves, that is, modulational stability, under localized small initial perturbations with nonlocalized initialmodulations.Lp(R)-nonlinear stability of suchwaves has been studied in Johnson et al. (2013) forp ≥ 2 by using Hausdorff-Young inequality. In this note, by using the pointwise estimates obtained in Jung, (2012) and Jung and Zumbrun (2016), we extend Lp(R)-nonlinear stability (p ≥ 2) in Johnson et al. (2013) to L1(R)-nonlinear stability. More precisely, we obtain L1(R)estimates of modulated perturbations ?̃?(x − ψ(x, t), t) − u(x) of u with a phase function ψ(x, t) under small initial perturbations consisting of localized initial perturbations ?̃?(x − h0(x), 0) − u(x) and nonlocalized initial modulations h0(x) = ψ(x, 0).


Introduction
Many evolutionary PDEs possess spatially periodic traveling waves and their stability has been widely studied in recent years.In this paper, by using the results of [1][2][3], we consider  1 (R)-nonlinear modulational stability of spatially periodic traveling waves in a system of reaction-diffusion equations: with  ∈ R,  ≥ 0, and  ∈ R  , where  : R  → R  is sufficiently smooth.Suppose that (, ) = ( − ) is a spatially 1-periodic traveling wave of (1) with a wave speed .By substituting (, ) = ( − ) into (1), one can say that () is a stationary 1-periodic solution of Indeed, () is a 1-periodic profile of traveling wave ODEs: Compared with other types of traveling wave solutions such as front or pulse, the main difficulty of study of stability of periodic traveling waves is that the linearized operator considered on the whole line has only essential spectrum; that is, the spectrum is continuous up to zero (see Section 1.1 for details).This gives no spectral gap from the origin.The spectral gap plays a very important role in the study of (linear and nonlinear) stability because it gives exponential decay of the linearized operator.This is the reason why stability of the periodic traveling waves has been an open problem for a long time.
However, in the late 1990s, nonlinear stability of bifurcating periodic traveling waves of Swift-Hohenberg equation with respect to the localized perturbation has been studied in [4,5] by using renormalization techniques.Stable diffusive mixing of periodic reaction-diffusion waves has been obtained in [6] based on a nonlinear decomposition of phase and amplitude variables and renormalization techniques.Johnson, Zumbrun, and their collaborators also showed   (R) ( ≥ 2) nonlinear modulational stability of periodic traveling waves of systems of reaction-diffusion equations and of conservation under both localized and nonlocalized perturbations in [1,[7][8][9].By using pointwise linear estimates together with a nonlinear iteration scheme developed by Johnson-Zumbrun, pointwise nonlinear stability of such waves has been also studied in [2,3,10].For other related works on modulated periodic traveling waves, we refer readers to [11][12][13].
To begin with, we first review the concept of stability of () of (2).Roughly speaking, we say that () is (boundedly) stable if any other solutions ũ(, ) of (2) which are initially near () stay near () for all  ≥ 0.More precisely, we understand the following (bounded) stability definition.Definition 1.Let  1 and  2 be Banach spaces and suppose that () is a stationary solution of nonlinear partial differential equation of the form where F includes all differential, linear, and nonlinear terms.
For 2 ≤  ≤ ∞,   (R)-estimates of such nonlocalized modulated perturbations have been already established in [1] and [9] for systems of reaction-diffusion equations and of conservation laws, respectively, by using the generalized Hausdorff-Young inequality for  ≤ 2 ≤  and 1/ + 1/ = 1, where ǔ (, ⋅) is a Bloch transform of  defined below in (12).This is the reason why their stability analysis has been restricted to  ≥ 2.

Preliminaries.
In order to study stability of , the spectral information of the linearization of (2) around  is required; so we first linearize the PDE (2) about ().In order to see the importance of linearization about , we consider again the general PDE of form (4).By setting perturbations of  as V(, ) = ũ(, ) − (), we have Here, V  = ()V is referred to as the linearization of (4) about ; so, from this linearization, we obtain the linear perturbation equation.We now linearize (2) around  and consider the eigenvalue problem of the form with 1-periodic coefficients.We consider this linear operator  on  2 (R) with densely defined domain  2 (R).In this section, we recall Bloch analysis [1,3,7] which is a key idea of spectral analysis of linear operators with periodic coefficients.If we apply Floquet theory [14] to the first-order ODE system obtained by (7),  ∈ C lies in  2 (R)-spectrum of  if and only if V has the form for some Floquet exponent  ∈ [−, ) and 1-periodic function  in .It means that, recalling the linear operator  acts on the whole line R, there is no  2 (R)-eigenfunction of ; so  2 (R)-spectrum of  introduced in ( 7) is entirely essential; that is, there is no isolated eigenvalue (point spectrum).That is why the spectrum is continuous up to zero without spectral gap.
Continuing with this setup, we recall the inverse Bloch-Fourier representation.Applying the inverse Fourier transform, we have, for any  ∈  2 (R), that so, for any  ∈  2 (R), where ǧ (, ) = ∑ ∈Z  2 ĝ( + 2) is referred to as the Bloch transform and ĝ(⋅) denotes the Fourier transform.Furthermore, noting that ǧ (, ) is 1-periodic in , (10) and (12) give us the formula of periodic coefficient solution operator   for : As a starting point, we used this formula to estimate linear behaviors of  in terms of the localized and nonlocalized data in [2] and [3], respectively.Indeed, pointwise estimates on Green function of , by plugging () into the Dirac delta function   (), have been obtained in [2] to estimate ()V 0 for a localized data V 0 .Moreover, in [3], we established the pointwise linear behavior of () on nonlocalized data ℎ 0 .

Spectral Stability.
With these preparations, following [4,5], we now state the spectral stability assumptions.
From (D1), we see that the origin is the only neutral spectrum of .By differentiating the traveling wave ODE (3), we obtain   () = 0; so 0 is an eigenvalue of  0 because   ∈  2 per ([0, 1]).The second assumption (D2) implies that, for sufficiently small ||, an eigenvalue () of   bifurcating from 0 is analytic in .If we use Taylor series expansion with respect to , by (D3) and the complex symmetry (−) = (), () can be written as for sufficiently small          , where  ∈ R and  > 0.Moreover, by the perturbation theory, the corresponding right and left eigenfunctions of   , denoted by (, ) and q(, ), respectively, are also analytic in  for sufficiently small ||.In particular, (D2) implies that one can take (, 0) =   () because  0   () = 0.Moreover, condition (D3) was verified by direct numerical Evans function analysis in [12].

Discussion and Open Problems.
As shown in Theorem 2,  is modulationally stable in  1 (R) under nonlocalized initial perturbations.That is, even if we perturb the underlying solution  by shifting it slightly out of phase at ±∞,  is still stable in  1 (R) with some modulation determined in Section 4.However, it is just boundedly stable, while the main theorem in [1] implies that the underlying periodic solution  is nonlinearly "asymptotically" stable in   (R) ( ≥ 2); that is, ũ(−(, ), ) converges to  in   (R) as  goes to infinity.It makes sense if we plug  = 1 into   -estimates in [1]: We emphasize again that we use pointwise estimates of the solution operator () in terms of the localized data V 0 fl ũ0 ( − ℎ 0 ()) − () and the nonlocalized data   ℎ 0 in order to obtain  1 (R)-stability, because we cannot apply the Hausdorff-Young inequality (5) as we mentioned in the Introduction.This is the reason why we need both initial conditions ( 16) and (17), while ( 16) is enough to prove   (R)stability ( ≥ 2) in [1].
Pointwise estimates give us more elaborate behaviors of the perturbation V like an exact solution of linear perturbation equation V  = V; so pointwise estimate is a common method to get  1 (R)-stability.However, we need to make more effort to obtain pointwise bounds.As shown in [3], we need condition (17) in order to obtain pointwise estimate of ()(  ℎ 0 )() for the case || ≫  with a sufficiently large constant  > 0. This issue did not arise in [2] because only localized modulations (, ) with ℎ 0 () = (, 0) ≡ 0 were considered in [2].However, the primary difficulty is to obtain pointwise estimates on the Green function (, ; ) of the linear operator  for the case | − | ≫  by using Bloch decomposition.If we solve the difficulty, we might delete condition (17) in Theorem 2, even in the main theorem of [3].This problem was discussed in more detail in [3].Pointwise nonlinear stability under nonlocalized perturbations in systems of conservation is an open problem.
The dynamics of modulated periodic traveling waves have been studied in [11] by the WKB approximations.We consider that the wave number of the periodic traveling waves is modulated by the function   .As described in Theorem 2 and [1,3],  is nonlinearly stable under small initial perturbations ( 16) and ( 17), with a heat kernel rate of decay in the wave number   .One can clearly see this in the integral representation of () in Section 4.

Review of Previous Results
This section provides the previous results in [2,3], particularly how to estimate the solution operator of  in terms of the localized and the nonlocalized data in pointwise sense.For some unknown modulation (, ), we first define modulated perturbations V of the underlying periodic solution  as for any solution ũ of (2) near .Recalling the linear operator  in (7), we review the nonlinear perturbation equation about V established in [1,7].
Lemma 3 (nonlinear perturbation equation).The modulated perturbation V satisfies where () is the solution operator of  and the formula of () is given by (13).The goal of this paper is to estimate (23) in  1 (; R) for an appropriate nonlocalized modulation (, ).The most important and difficult part is estimating pointwise bounds of () with respect to the localized data V 0 and the nonlocalized data   ℎ 0 .For the localized data V 0 , we use pointwise bounds on Green's function (, ; ) of  in [2]  (, ; ) =   ()  (, ; ) + G (, ; ) , uniformly on  ≥ 0, for some sufficiently large constants  > 0 and  > 0.Here q is the periodic left eigenfunction of  0 at  = 0 discussed in Section 1.2 and () is a smooth cutoff function such that () = 0 for 0 ≤  ≤ 1/2 and () = 1 for  ≥ 1.
For the nonlocalized data   ℎ 0 , we decompose the solution operator () as follows: Here, ̂denotes the Fourier transform.This kind of decomposition was not needed in [2] because we considered only localized modulations (, ) with ℎ 0 () = 0.Moreover, the decomposition of () here is rather different from the decomposition in [1] because we need pointwise estimates of S * ()(  ℎ 0 ) obtained in [3] in order to prove  1 (R)-stability.